More on equal-area histograms

Today, I'm returning to those "equal-area histograms" that Andrew wrote about last month. I have two previous posts about this. The first post introduces the concept: in a traditional histogram, the columns have the same bin width while the column heights can represent a variety of metrics, such as counts, relative frequencies (i.e. proportion of the data) and densities; in the equal-area histogram, the columns have varying widths while the area of each column is constant, and determined by the number of bins (columns).

Here is a comparison of the two types of histograms.

In a second post, I explained the differences between using counts, frequencies and densities in the vertical axis. The underlying issue is that the histogram is not merely a column chart, in which the width of the columns is arbitrary and data-free - in the histogram, both the heights and widths of columns carry meaning. One feature of the histogram that almost everyone expects is that the area of the columns sum up to 1. This aligns with a desired interpretation of probabilities of data falling into specified ranges, as we'd like the amount of data in the entire range to add up to 100%. Unfortunately, the two items are usually incompatible with each other.

If the height of the columns represents the probability of data falling into the range as indicated by its width, then the sum of the column heights is 1, which implies that the sum of the column areas cannot be 1. On the other hand, if the column areas add up to 1, then the column heights will not add up to 1, and thus, in this scenario, we cannot interpret the column heights to be probabilities. As explained in the second post, the column heights in this situation are densities, which can be defined as the proportion of data divided by the bin width. Intuitively, it gives information on how dense or sparse the data are within the specified range.

***

Today's post start with a toy dataset, containing randomly generated values from a normal distribution (bell curve) centered at 4 and with standard deviation 1.

Here is the traditional histogram of the dataset, using 100 equal-width bin. (I generated 10,000 values)

Next, I created a panel of four equal-area histograms, with increasingly number of bins. Each is built from the same underlying dataset.

The first histogram divides the data into 4 bins; then 10 bins, 20 bins and 100 bins.

In the 4-bin case, each column contains 1/4 = 25% of the data. The middle two columns contain 50% of the data, and they have high densities, as the widths of these columns are low. It's a crude approximation of the familiar bell curve.

As we increase the number of bins, the columns in the middle of the distribution, where most of the data are concentrated, become narrower. In the sparse regions, the column width doesn't necessarily grow because each column must contain 1/n of the data, where n is the number of columns. As the number of columns increases, each column contains less of the data.

The bottom chart is the "percentogram", which is what Andrew's correspondent proposed. The number of bins is set to 100, so each column contains exactly 1 percent of the data. For a normal distribution, the columns in the middle are very tall and thin.

The reason why the middle of the percentogram looks faded is that I asked for a white border around each column. But when the columns are so thin, even if one sets the border width very small, what readers see is a mixture of orange and white.

With high number of bins, we notice a few things: a) the outline of the histogram becomes "ragged" (the more bins there are), b) the middle columns become razor-thin c) the width conceded by the middle columns is absorbed not by the columns at the edges but those between the peak and the edge.

I'm struggling a bit to justify this percentogram versus the typical, equal-width histogram.

Let me go down a different path.

***

In "principled" histograms, the column heights represent data densities, while the total area of the columns add up to 1. This leads us to a new understanding of the relationship between the equal-width histogram and the equal-area histogram.

We start with data density defined by (proportion of data) / (bin width). Those two values are not independent - one is fully determined by the other, given the underlying dataset. In a traditional equal-width histogram, the question is: how much of the data is found in a column of fixed width? In the new equal-area histogram, the question is: how wide is the bin that contains a fixed amount of data? In the former, the denominator is fixed while the numerator varies; the opposite occurs in the latter.

***

We also recognize that given the range of the data, there is a relationship between the the set of bin widths in the two types of histograms. In the traditional histogram, all bin widths have the same value, equal to the range of the data divided by the number of bins. Think of this as the average bin width. In an equal-area histogram, the set of bin widths varies: however, the sum of the bin widths must still add up to the range of the data. For two comparable histograms with the same number of bins, the average of the bin widths must be the same for both sets. (I'm ignoring any rounding situations in which the range of the histogram is larger than the range of the data.)

Now, consider the middle of the normal distribution where the data are dense. In the traditional histogram, the column in the middle still has width equal to the average bin width. In the equal-area histogram, the middle column has width much smaller than the average bin width. In other words, we can think of the column in the traditional histogram being broken up into many thin and slim columns in the equal-area histogram, each containing 1% of the data in the case of the percentogram.

The height of the column is the data density. In the traditional histogram, the middle column is the pooled sample of larger size; in the equal-area histogram, each of those thin and slim columns is a partition of the sample. This explains observation (a) above in which the outline of the equal-area histogram is more ragged - it's because each column contains fewer data from which to estimate the data density.

But this raggedness is artificial, sampling noise.

***

The sparse areas are more complicated still. It's also the reverse of the above. On the edges of the normal distribution, the columns of the new histogram are wider than those of the traditional histogram. So, we can think of breaking up the edge column of the new histogram into multiple columns of the traditional histogram.

The interpretation is more complicated because the data are sparse in this region. Obviously, the estimates of density on the traditional histogram in sparse regions are poor because not enough data reside in there. The density estimate on the new histogram is based on a larger sample size.

However.

Yes, however, whether the new histogram's density estimate is better depends on the shape of the tail of the distribution. A normal distribution has exponential tails, which means that the data density declines quite drastically the further we go into the tail. Therefore, the new histogram averages the data densities across a large part of the tail, wiping out the exponential shape while the traditional histogram preserves that shape - at the expense of greater sampling variability due to smaller sample sizes.

***

For what it's worth, let's look at some histograms for an exponential random variable.

Here is the traditional histogram:

The data are extremely dense on the left side while it has a long tail on the right side.

Here are the four equal-area histograms for 4, 10, 20 and 100 bins.

The four-bin version gives a nice summary of the shape. As the number of bins goes up, as before, the denser regions now have tall, thin spikes. Again, because of the white borders, the last histogram with 100 bins is faded where the data are densest. (So obviously, don't follow my lead, and eliminate borders if you want to use it.)

The 100-bin version looks almost the same as the traditional histogram.

***

At this stage of the exploration, I still haven't found a compelling reason to switch to equal-area hist0grams. In the denser regions, it's adding sampling noise. If I don't care about the sparser areas, specifically, the shape of the tails, maybe they provide a cleaner presentation.

 

Visual story-telling: do you know or do you think?

One of the most important data questions of all time is: do you know? or do you think?

And one of the easiest traps to fall into is: I think, therefore I know.

***

Visual story-telling can be great but it can also mislead. Deception sometimes happens when readers are nudged to "fill in the blanks" with stuff they think they know, but they don't.

A Twitter reader asked me to look at the map in this Los Angeles Times (paywall) opinion column.

The column promptly announces its premise:

Years of widening economic inequality, compounded by the pandemic and political storm and stress, have given Americans the impression that the country is on the wrong track. Now there’s empirical data to show just how far the country has run off the rails: Life expectancies have been falling.

The writer creates the expectation that he will reveal evidence in the form of data to show that life expectancies have been driven down by economic inequality, pandemic, and politics. Does he succeed?

***

The map portrays average life expectancy (at birth) for some mysterious, presumably very recent, year for every county in the United States. From the color legend, we learn that the bottom-to-top range is about 20 years. There is a clear spatial pattern, with the worst results in the south (excepting south Florida).

The choice of colors is telling. Red and blue on a U.S. map has heavy baggage, as they signify the two main political parties in the country. Given that the author believes politics to be a key driver of health outcomes, the usage of red and blue here is deliberate. Throughout the article, the columnist connects the lower life expectancies in southern states to its politics.

For example, he said "these geographical disparities aren't artifacts of pure geography or demographics; they're the consequences of policy decisions at the state level... Of the 20 states with the worst life expectancies, eight are among the 12 that have not implemented Medicaid expansion under the Affordable Care Act..."

Casual readers may fall into a trap here. There is nothing on the map itself that draws the connection between politics and life expectancies; the idea is evoked purely through the red-blue color scheme. So, as readers, we are filling in the blanks with our own politics.

What could have been done instead? Let's look at the life expectancy map side by side with the map of the U.S. 2020 Presidential election.

Because of how close recent elections have been, we may think the political map has a nice balance of red and blue but it isn't. The Democrats' votes are heavily concentrated in densely-populated cities so most of the Presidential election map is red. When placed next to each other, it's obvious that politics don't explain the variance in life expectancy well. The Midwest is deep red and yet they have above average life expectancies. I have circled out various regions that contradict the claim that Republican politics drove life expectancies down.

It's not sufficient to point to the South, in which Republican votes and life expectancy are indeed inversely correlated. A good theory has to explain most of the country.

***

The columnist also suggests that poverty is the cause of low life expectancy. That too cannot be gleaned from the published map. Again, readers are nudged to use their wild imagination to fill in the blank.

Data come to the rescue. Here is a side-by-side comparison of the map of life expectancies and the map of median incomes.

A similar conundrum. While the story feels right in the South, it fails to explain the northwest, Florida, and various other parts of the country. Take a look again at the circled areas. Lower income brackets are also sometimes associated with high life expectancies.

***

The author supplies a third cause of lower life expectancies: Covid-19 response. Because Covid-19 was the "most obvious and convenient" explanation for the loss of life expectancy during the pandemic, this theory suggests that the red areas on the life expectancy map should correspond to the regions most ravaged by Covid-19.

Let's see the data.

The map on the right shows the number of confirmed cases until June 2021. As before, the correlation holds somewhat in the South but there are notable exceptions, e.g. the Midwest. We also have states with low Covid-19 cases but below-average life expectancy.

***

What caused the decline of life expectancy in the U.S. - which began before the pandemic, and has continued beyond - is highly complex, beyond what a single map or a pair of maps or a few pairs of maps could convey. Showing a red-blue map presents a trap for readers to fall into, in which they start thinking, without knowing.

 

The one thing you're afraid to ask about histograms

In the previous post about a variant of the histogram, I glossed over a few perplexing issues - deliberately. Today's post addresses one of these topics: what is going on in the vertical axis of a histogram?

The real question is: what data are encoded in the histogram, and where?

***

Let's return to the dataset from the last post. I grabbed data from a set of international football (i.e. soccer) matches. Each goal scored has a scoring minute. If the goal is scored in regulation time, the scoring minute is a number between 1 and 90 minutes. Specifically, the data collector applies a rounding up: any goal scored between 0 and 60 seconds is recorded as 1, all the way up to a goal scored between 89 and 90th minute being recorded as 90. In this post, I only consider goals scored in regulation time so the horizontal axis is between 1-90 minutes.

The kneejerk answer to the posed question is: counts in bins. Isn't it the case that in constructing a histogram, we divide the range of values (1-90) into bins, and then plot the counts within bins, i.e. the number of goals scored within each bin of minutes?

The following is what we have in mind:

Let's call this the "count histogram".

Some readers may dislike the scale of the vertical axis, as its interpretation hinges on the total sample size. Hence, another kneejerk answer is: frequencies in bins. Instead of plotting counts directly, plot frequencies, which are just standardized counts. Just divide each value by the sample size. Here's the "frequency histogram":

The count and frequency histograms are identical except for the scale, and appear intuitively clear. The count and frequency data are encoded in the heights of the columns. The column widths are an afterthought, and they adhere to a fixed constant. Unlike a column chart, typically the gap width in a histogram is zero, as we want to partition the horizontal range into adjoining sections.

Now, if you look carefully at the histogram from the last post, reproduced below, you'd find that it plots neither counts nor frequencies:

The numbers on the axis are fractions, and suggest that they may be frequencies, but a quick check proves otherwise: with 9 columns, the average column should contain at least 10 percent of the data. The total of the displayed fractions is nowhere near 100%, which is our expectation if the values are relative frequencies. You may have come across this strangeness when creating histograms using R or some other software.

The purpose of this post is to explain what values are being plotted and why.

***

What are the kinds of questions we like to answer about the distribution of data?

At a high level, we want to know "where are my data"?

Arguably these two questions are fundamental:

• what is the probability that the data falls within a given range of values? e.g., what is the probability that a goal is scored in the first 15 minutes of a football match?
• what is the relative probability of data between two ranges of values? e.g. are teams more likely to score in last 5 minutes of the first half or the last five minutes of the second half of a football match?

In a histogram, the first question is answered by comparing a given column to the entire set of columns while the second question is answered by comparing one column to another column.

Let's see what we can learn from the count histogram.

In a count histogram, the heights encode the count data. To address the relative probability question, we note that the ratio of heights is the ratio of counts, and the ratio of counts is the same as the ratio of frequencies. Thus, we learn that teams are roughly 3000/1500 = 1.5 times more likely to score in the last 5 minutes of the second half than during the last 5 minutes of the first half. (See the green columns).

[For those who follow football, it's clear that the data collector treated goals scored during injury time of either half as scored during the last minute of the half, so this dataset can't be used to analyze timing of goals unless the real minutes were recorded for injury-time goals.]

To address the range probability question, we compare the aggregate height of the three orange columns with the total heights of all columns. Note that I said "height", not "area," because the heights directly encode counts. It's actually taxing to figure out the total height!

We resort to reading the total area of all columns. This should yield the correct answer: the area is directly proportional to the height because the column widths are fixed as a constant. Bear in mind, though, if the column widths vary (the theme of the last post), then areas and heights are not interchangable concepts.

Estimating the total area is still not easy, especially if the column heights exhibit high variance. What we need is the proportion of the total area that is orange. It's possible to see, not easy.

You may interject now to point out that the total area should equal the aggregate count (sample size). But that is a fallacy! It's very easy to make this error. The aggregate count is actually the total height, and because of that, the total area is the aggregate count multiplied by the column width! In my example, the total height is 23,682, which is the number of goals in the dataset, while the total area is 23,682 times 5 minutes.

[For those who think in equations, the total area is the sum over all columns of height(i) x width(i). When width is constant, we can take it outside the sum, and the sum of height(i) is just the total count.]

***

The count histogram is hard to use because it requires knowing the sample size. It's the first thing that is produced because the raw data are counts in bins. The frequency histogram is better at delivering answers.

In the frequency histogram, the heights encode frequency data. We can therefore just read off the relative probability of the orange column, bypassing the need to compute the total area.

This workaround actually promotes the fallacy described above for the count histogram. It is easy to fall into the trap of thinking that the total area of all columns is 100%. It isn't.

Similar to before, the total height should be the total frequency but the total area is the total frequency multipled by the column width, that is to say, the total area is the reciprocal of the bin width. In the football example, using 5-minute intervals, the total area of the frequency histogram is 1/(5 minutes) in the case of equal bin widths.

How about the relative probability question? On the frequency histogram, the ratio of column heights is the ratio of frequencies, which is exactly what we want. So long as the column width is constant, comparing column heights is easy.

***

One theme in the above discussion is that in the count and frequency histograms, the count and frequency data are encoded in the column heights but not the column areas. This is a source of major confusion. Because of the convention of using equal column widths, one treats areas and heights as interchangable... but not always. The total column area isn't the same as the total column height.

This observation has some unsettling implications.

As shown above, the total area is affected by the column width. The column width in an equal-width histogram is the range of the x-values divided by the number of bins. Thus, the total area is a function of the number of bins.

Consider the following frequency histograms of the same scoring minutes dataset. The only difference is the number of bins used.

Increasing the number of bins has a series of effects:

• the columns become narrower
• the columns become shorter, because each narrower bin can contain at most the same count as the wider bin that contains it.
• the total area of the columns become smaller.

This last one is unexpected and completely messes up our intuition. When we increase the number of bins, not only are the columns shortening but the total area covered by all the columns is also shrinking. Remember that the total area whether it is a count or frequency histogram has a factor equal to the bin width. Higher number of bins means smaller bin width, which means smaller total area.

***

What if we force the total area to be constant regardless of how many bins we use? This setting seems more intuitive: in the 5-bin histogram, we partition the total area into five parts while in the 10-bin histogram, we divide it into 10 parts.

This is the principle used by R and the other statistical software when they produce so-called density histograms. The count and frequency data are encoded in the column areas - by implication, the same data could not have been encoded simultaneously in the column heights!

The way to accomplish this is to divide by the bin width. If you look at the total area formulas above, for the count histogram, total area is total count x bin width. If the height is count divided by bin width, then the total area is the total count. Similarly, if the height in the frequency histogram is frequency divided by bin width, then the total area is 100%.

Count divided by some section of the x-range is otherwise known as "density". It captures the concept of how tightly the data are packed inside a particular section of the dataset. Thus, in a count-density histogram, the heights encode densities while the areas encode counts. In this case, total area is the total count. If we want to standardize total area to be 1, then we should compute densities using frequencies rather than counts. Frequency densities are just count densities divided by the total count.

To summarize, in a frequency-density histogram, the heights encode densities, defined as frequency divided by the bin width. This is not very intuitive; just think of densities as how closely packed the data are in the specified bin. The column areas encode frequencies so that the total area is 100%.

The reason why density histograms are confusing is that we are reading off column heights while thinking that the total area should add up to 100%. Column heights and column areas cannot both add up to 100%. We have to pick one or the other.

Comparing relative column heights still works when the density histogram has equal bin widths. In this case, the relative height and relative area are the same because relative density equals relative frequencies if the bin width is fixed.

The following charts recap the discussion above. It shows how the frequency histogram does not preserve the total area when bin sizes are changed while the density histogram does.

***

The density histogram is a major pain for solving range probability questions because the frequencies are encoded in the column areas, not the heights. Areas are not marked out in a graph.

The column height gives us densities which are not probabilities. In order to retrieve probabilities, we have to multiply the density by the bin width, that is to say, we must estimate the area of the column. That requires mapping two dimensions (width, height) onto one (area). It is in fact impossible without measurement - unless we make the bin widths constant.

When we make the bin widths constant, we still can't read densities off the vertical axis, and treat them as probabilities. If I must use the density histogram to answer the question of how likely a team scores in the first 15 minutes, I'd sum the heights of the first 3 columns, which is about 0.025, and then multiply it by the bin width of 5 minutes, which gives 0.125 or 12.5%.

At the end of this exploration, I like the frequency histogram best. The density histogram is useful when we are comparing different histograms, which isn't the most common use case.

***

The histogram is a basic chart in the tool kit. It's more complicated than it seems. I haven't come across any intro dataviz books that explain this clearly.

Most of this post deals with equal-width histograms. If we allow bin widths to vary, it gets even more complicated. Stay tuned.

***

For those using base R graphics, I hope this post helps you interpret what they say in the manual. The default behavior of the "hist" function depends on whether the bins are equal width:

• if the bin width is constant, then R produces a count histogram. As shown above, in a count histogram, the column heights indicate counts in bins but the total column area does not equal the total sample size, but the total sample size multiplied by the bin width. (Equal width is the default unless the user specifies bin breakpoints.)
• if the bin width is not constant, then R produces a (frequency-)density histogram. The column heights are densities, defined as frequencies divided by bin width while the column areas are frequencies, with the total area summing to 100%.

Unfortunately, R does not generate a frequency histogram. To make one, you'd have to divide the counts in bins by the sum of counts. (In making some of the graphs above, I tricked it.) You also need to trick it to make a frequency-density histogram with equal-width bins, as it's coded to produce a count histogram when bin size is fixed.

 

P.S. [5-2-2023] As pointed out by a reader, I should clarify that R and I use the word "frequency" differently. Specifically, R uses frequency to mean counts, therefore, what I have been calling the "count histogram", R would have called it a "frequency histogram", and what I have been describing as a "frequency histogram", the "hist" function simply does not generate it unless you trick it to do so. I'm using "frequency" in the everyday sense of the word, such as "the frequency of the bus". In many statistical packages, frequency is used to mean "count", as in the frequency table which is just a table of counts. The reader suggested proportion which I like, or something like weight.

 

 

 

 

 

Bivariate choropleths

A reader submitted a link to Joshua Stephen's post about bivariate choropleths, which is the technical term for the map that FiveThirtyEight printed on abortion bans, discussed here. Joshua advocates greater usage of maps with two-dimensional color scales.

As a reminder, the fundamental building block is expressed in this bivariate color legend:

Counties are classified into one of these nine groups, based on low/middle/high ratings on two dimensions, distance and congestion.

The nine groups are given nine colors, built from superimposing shades of green and pink. All nine colors are printed on the same map.

Without a doubt, using these nine related colors are better than nine arbitrary colors. But is this a good data visualization?

Specifically, is the above map better than the pair of maps below?

The split map is produced by Josh to explain that the bivariate choropleth is just the superposition of two univariate choropleths. I much prefer the split map to the superimposed one.

***

Think about what the reader goes through when comparing two counties.

Superimposing the two univariate maps solves one problem: it removes the need to scan back and forth between two maps, looking for the same locations, something that is imprecise. (Unless, the map is interactive, and highlighting one county highlights the same county in the other map.)

For me, that's a small price to pay for quicker translation of color into information.

 

 

Finding the story in complex datasets

In CT Mirror's feature about Connecticut, which I wrote about in the previous post, there is one graphic that did not rise to the same level as the others.

This section deals with graduation rates of the state's high school districts. The above chart focuses on exactly five districts. The line charts are organized in a stack. No year labels are provided. The time window is 11 years from 2010 to 2021. The column of numbers show the difference in graduation rates over the entire time window.

The five lines look basically the same, if we ignore what looks to be noisy year-to-year fluctuations. This is due to the weird aspect ratio imposed by stacking.

Why are those five districts chosen? Upon investigation, we learn that these are the five districts with the biggest improvement in graduation rates during the 11-year time window.

The same five schools also had some of the lowest graduation rates at the start of the analysis window (2010). This must be so because if a school graduated 90% of its class in 2010, it would be mathematically impossible for it to attain a 35% percent point improvement! This is a dissatisfactory feature of the dataviz.

***

In preparing an alternative version, I start by imagining how readers might want to utilize a visualization of this dataset. I assume that the readers may have certain school(s) they are particularly invested in, and want to see its/their graduation performance over these 11 years.

How does having the entire dataset help? For one thing, it provides context. What kind of context is relevant? As discussed above, it's futile to compare a school at the top of the ranking to one that is near the bottom. So I created groups of schools. Each school is compared to other schools that had comparable graduation rates at the start of the analysis period.

Amistad School District, which takes pole position in the original dataviz, graduated only 58% of its pupils in 2010 but vastly improved its graduation rate by 35% over the decade. In the chart below (left panel), I plotted all of the schools that had graduation rates between 50 and 74% in 2010. The chart shows that while Amistad is a standout, almost all schools in this group experienced steady improvements. (Whether this phenomenon represents true improvement, or just grade inflation, we can't tell from this dataset alone.)

The right panel shows the group of schools with the next higher level of graduation rates in 2010. This group of schools too increased their graduation rates almost always. The rate of improvement in this group is lower than in the previous group of schools.

The next set of charts show school districts that already achieved excellent graduation rates (over 85%) by 2010. The most interesting group of schools consists of those with 85-89% rates in 2010. Their performance in 2021 is the most unpredictable of all the school groups. The majority of districts did even better while others regressed.

Overall, there is less variability than I'd expect in the top two school groups. They generally appeared to have been able to raise or maintain their already-high graduation rates. (Note that the scale of each chart is different, and many of the lines in the second set of charts are moving within a few percentages.)

One more note about the charts: The trend lines are "smoothed" to focus on the trends rather than the year to year variability. Because of smoothing, there is some awkward-looking imprecision e.g. the end-to-end differences read from the curves versus the observed differences in the data. These discrepancies can easily be fixed if these charts were to be published.

Thoughts on Daniel's fix for dual-axes charts

I've taken a little time to ponder Daniel Z's proposed "fix" for dual-axes charts (link). The example he used is this:

In that long post, Daniel explained why he preferred to mix a line with columns, rather than using the more common dual lines construction: to prevent readers from falsely attributing meaning to crisscrossing lines. There are many issues with dual-axes charts, which I won't repeat in this post; one of their most dissatisfying features is the lack of connection between the two vertical scales, and thus, it's pretty easy to manufacture an image of correlation when it doesn't exist. As shown in this old post, one can expand or restrict one of the vertical axes and shift the line up and down to "match" the other vertical axis.

Daniel's proposed fix retains the dual axes, and he even restores the dual lines construction.

How is this chart different from the typical dual-axes chart, like the first graph in this post?

Recall that the problem with using two axes is that the designer could squeeze, expand or shift one of the axes in any number of ways to manufacture many realities. What Daniel effectively did here is selecting one specific way to transform the "New Customers" axis (shown in gray).

His idea is to run a simple linear regression between the two time series. Think of fitting a "trendline" in Excel between Revenues and New Customers. Then, use the resulting regression equation to compute an "estimated" revenues based on the New Customers series. The coefficients of this regression equation then determines the degree of squeezing/expansion and shifting applied to the New Customers axis.

The main advantage of this "fix" is to eliminate the freedom to manufacture multiple realities. There is exactly one way to transform the New Customers axis.

The chart itself takes a bit of time to get used to. The actual values plotted in the gray line are "estimated revenues" from the regression model, thus the blue axis values on the left apply to the gray line as well. The gray axis shows the respective customer values. Because we performed a linear fit, each value of estimated revenues correspond to a particular customer value. The gray line is thus a squeezed/expanded/shifted replica of the New Customers line (shown in orange in the first graph). The gray line can then be interpreted on two connected scales, and both the blue and gray labels are relevant.

***

What are we staring at?

The blue line shows the observed revenues while the gray line displays the estimated revenues (predicted by the regression line). Thus, the vertical gaps between the two lines are the "residuals" of the regression model, i.e. the estimation errors. If you have studied Statistics 101, you may remember that the residuals are the components that make up the R-squared, which measures the quality of fit of the regression model. R-squared is the square of r, which stands for the correlation between Customers and the observed revenues. Thus the higher the (linear) correlation between the two time series, the higher the R-squared, the better the regression fit, the smaller the gaps between the two lines.

***

There is some value to this chart, although it'd be challenging to explain to someone who has not taken Statistics 101.

While I like that this linear regression approach is "principled", I wonder why this transformation should be preferred to all others. I don't have an answer to this question yet.

***

Daniel's fix reminds me of a different, but very common, chart.

This chart shows actual vs forecasted inflation rates. This chart has two lines but only needs one axis since both lines represent inflation rates in the same range.

We can think of the "estimated revenues" line above as forecasted or expected revenues, based on the actual number of new customers. In particular, this forecast is based on a specific model: one that assumes that revenues is linearly related to the number of new customers. The "residuals" are forecasting errors.

In this sense, I think Daniel's solution amounts to rephrasing the question of the chart from "how closely are revenues and new customers correlated?" to "given the trend in new customers, are we over- or under-performing on revenues?"

Instead of using the dual-axes chart with two different scales, I'd prefer to answer the question by showing this expected vs actual revenues chart with one scale.

This does not eliminate the question about the "principle" behind the estimated revenues, but it makes clear that the challenge is to justify why revenues is a linear function of new customers, and no other variables.

Unlike the dual-axes chart, the actual vs forecasted chart is independent of the forecasting method. One can produce forecasted revenues based on a complicated function of new customers, existing customers, and any other factors. A different model just changes the shape of the forecasted revenues line. We still have two comparable lines on one scale.

 

 

 

 

 

All about Connecticut

This dataviz project by CT Mirror is excellent. The project walks through key statistics of the state of Connecticut.

Here are a few charts I enjoyed.

The first one shows the industries employing the most CT residents. The left and right arrows are perfect, much better than the usual dot plots.

The industries are sorted by decreasing size from top to bottom, based on employment in 2019. The chosen scale is absolute, showing the number of employees. The relative change is shown next to the arrow heads in percentages.

The inclusion of both absolute and relative scales may be a source of confusion as the lengths of the arrows encode the absolute differences, not the relative differences indicated by the data labels. This type of decision is always difficult for the designer. Selecting one of the two scales may improve clarity but induce loss aversion.

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The next example is a bumps chart showing the growth in residents with at least a bachelor's degree.

This is more like a slopegraph as it appears to draw straight lines between two time points 9 years apart, omitting the intervening years. Each line represents a state. Connecticut's line is shown in red. The message is clear. Connecticut is among the most highly educated out of the 50 states. It maintained this advantage throughout the period.

I'd prefer to use solid lines for the background states, and the axis labels can be sparser.

It's a little odd that pretty much every line has the same slope. I'm suspecting that the numbers came out of a regression model, with varying slopes by state, but the inter-state variance is low.

In the online presentation, one can click on each line to see the values.

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The final example is a two-sided bar chart:

This shows migration in and out of the state. The red bars represent the number of people who moved out, while the green bars represent those who moved into the state. The states are arranged from the most number of in-migrants to the least.

I have clipped the bottom of the chart as it extends to 50 states, and the bottom half is barely visible since the absolute numbers are so small.

I'd suggest showing the top 10 states. Then group the rest of the states by region, and plot them as regions. This change makes the chart more compact, as well as more useful.

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There are many other charts, and I encourage you to visit and support this data journalism.

 

 

 

Yet another off radar plot 2

In the last post, I described my experience reading the radar plot, by Bloomberg Graphics, that compares countries in terms of their citizens' post-retirement lives.

I used a different approach:

Instead of focusing on the actual time points (ages), my chart highlights the variance from the OECD averages.

The chart compares countries along three metrics: total life expectancy (including healthy and unhealthy periods), effective retirement age, and the number of healthy years in retirement, which is the issue of greatest interest.

From the above chart, France and Luxembourg have the same profiles. Their citizens live a year or two above the average life expectancy. They retire about 5 years earlier than average, and enjoy about 5 more years of healthy retirement.

Meanwhile, the life expectancy of Americans is about the same as the average OECD resident. Retirement also occurs around the same age as the OECD average. Nevertheless, Americans end up with fewer years of healthy retirement than the OECD average.

 

 

Area chart is not the solution

A reader left a link to a Wiki chart, which is ghastly:

This chart concerns the trend of relative proportions of House representatives in the U.S. Congress by state, and can be found at this Wikipedia entry. The U.S. House is composed of Representatives, and the number of representatives is roughly proportional to each state's population. This scheme actually gives small states disporportional representation, since the lowest number of representatives is 1 while the total number of representatives is fixed at 435.

We can do a quick calculation: 1/435 = 0.23% so any state that has less than 0.23% of the population is over-represented in the House. Alaska, Vermont and Wyoming are all close to that level. The primary way in which small states get larger representation is via the Senate, which sits two senators per state no matter the size. (If you've wondered about Nate Silver's website: 435 Representatives + 100 Senators + 3 for DC = 538 electoral votes for U.S. Presidental elections.)

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So many things have gone wrong with this chart. There are 50 colors for 50 states. The legend arranges the states by the appropriate metric (good) but in ascending order (bad). This is a stacked area chart, which makes it very hard to figure out the values other than the few at the bottom of the chart.

A nice way to plot this data is a tile map with line charts. I found a nice example that my friend Xan put together in 2018:

A tile map is a conceptual representation of the U.S. map in which each state is represented by equal-sized squares. The coordinates of the states are distorted in order to line up the tiles. A tile map is a small-multiples setup in which each square contains a chart of the same design to faciliate inter-state comparisons.

In the above map, Xan also takes advantage of the foregrounding concept. Each chart actually contains all 50 lines for every state, all shown in gray while the line for the specific state is bolded and shown in red.

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A chart with 50 lines looks very different from one with 50 areas stacked on each other. California, the most populous state, has 12% of the total population so the line chart has 50 lines that will look like spaghetti. Thus, the fore/backgrounding is important to make sure it's readable.

I suspect that the designer chose a stacked area chart because the line chart looked like spaghetti. But that's the wrong solution. While the lines no longer overlap each other, it is a real challenge to figure out the state-level trends - one has to focus on the heights of the areas, rather than the boundary lines.

[P.S. 2/27/2023] As we like to say, a picture is worth a thousand words. Twitter reader with the handle LHZGJG made the tile map I described above. It looks like this:

You can pick out the states with the key changes really fast. California, Texas, Florida on the upswing, and New York, Pennsylvania going down. I like the fact that the state names are spelled out. Little tweaks are possible but this is a great starting point. Thanks LHZGJG! ]

 

Getting simple charts right

Ian K. submitted this chart on Twitter:

The chart comes from a video embedded in this report (link) about Chicago cops leaving their jobs.

Let's start with the basics. This is an example of a simple line chart illustrating a time series of five observations. The vertical axis starts at 10,000 instead of 0. With this choice, the designer wants to focus on the point-to-point change in values, rather than its relation to the initial value.

Every graph has add-ons that assist cognition. On this chart, we have axis labels, gridlines and data labels. Every add-on increases reading time so we should be sparing.

First consider the gridlines. In the following chart, I conduct a self-sufficiency test by removing the data labels from the chart:

You can see that the last three values present no problems. The first two, especially the first value, are hard to read - because the top gridline is missing! The next chart restores the bounding gridline, so you can see the difference that one small detail can make:

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Next, let's compare the following versions of the chart. The left one contains data labels without gridlines and axis labels. The right one has the gridlines and axis labels but no data labels.

The left chart prints the entire dataset onto the chart. The reader in essence is reading the raw data. That appears to be the intention of the chart designer as the data labels are in large size, placed inside shiny white boxes. The level of the boxes determines the reader's perception as those catch more of our attention than the dots that actually represent the data.

The right chart highlights the dots and the lines between them. The gridlines are way too thick and heavy so as to distract rather than abet. This chart presumes that the reader isn't that interested in the precise numbers as she is in the trend.

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As Ian pointed out, one of the biggest problems with this chart is the appearance of even time intervals when all except one of the date values are January. This seemingly innocent detail destroys the chart. The line segments of the chart encodes the pre-post change in the staffing numbers. For most of the line segments, the metric is year-on-year change but the last two line segments on the right show something else: a 19-month change, followed by a 5-month change.

I did the following analysis to understand how big of a staffing problem CPD faces.

First I restored the January 2022 time value, while shifting the Aug 2022 value to its rightful place on the time axis. Next, I added the dashed brown line, which represents a linear extension of the trend seen between January 2020-2021, before the sudden dip. We don't know what the true January 2022 value is but the projected value based on past trend is around 12,200. By August, the projected value is around 11,923, about 300 above the actual value of 11,611. By January 2023, the projected value is almost exactly the same as the actual value.

This linear trending analysis is likely too simplistic but it offers a baseline to start thinking about what the story is. The long-term trend is still down but the apparent dip in 2022 may not be meaningful.